It's good that you have asked for help this early then, it will give you two days to finish it.

So a good first try is always to check if you can't just plug in the numbers.
For example
[tex]\lim_{x \to 0} \frac{x^2 + 7x - 8}{\sqrt{x^3} - 3 \sin(x) + 7 \cos(x) + 1}[/tex]
looks terrible, but plugging in x = 0 shows that the limit is -1.

any methods or working will do.. and i've totally no idea how to do it..
Plus my assignment is due this Wed..

You need to be sure you know what l'Hopital's rule is before you use it, as well as being absolutely sure that it is valid on your assignment. If you are sure you can use it, then both (a) and (b) can be done using l'Hopital's rule.

for b) try writing out the modulus as a normal function as it behaves in a region around x = 0. This shouldn;t be an issue as the modulus only has a kink where the argument changes from poistive to negative

with arguments 2x-1 and 2x+1, this will only be around the points x=1/2 and x= -1/2

for c) as a start, i would try expanding the expressions in terms of angle sum and double angle formula, to see if that helps